Download 08-Water_Chemistry-A..

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We’ve all heard of them; Why do they exist; how
are they computed
 A work term – the amount of energy needed relative to
a reference, or starting point
1000 kg
Standard
Energy Cost
Real Energy
Cost
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
The Physical Chemist’s secret: the activity coefficient
 A numerical correction applied to a model to predict
properties when systems do not behave ideally

The ultimate fudge factor
 An Activity Coefficients is a mathematical correction that
describes the effects of the immediate environment on a
dissolved ion.
 The numerical value is the quantitative deviation from
ideality - a value of 1 is ideal

What are the causes
 Ideally -Ions behave independently, in dilute solutions
because they are relatively far apart
 Nonideally - At higher concentrations, an ion’s electric field
affects other ions and changes their behavior
*
State 2
State 1
Work
+
Pure water
Pure water
𝑝𝑢𝑟𝑒 𝑤𝑎𝑡𝑒𝑟
𝐺𝑆𝑡𝑎𝑡𝑒2
*
𝑝𝑢𝑟𝑒 𝑤𝑎𝑡𝑒𝑟
− 𝐺𝑆𝑡𝑎𝑡𝑒1
𝑝𝑢𝑟𝑒 𝑤𝑎𝑡𝑒𝑟
= ∆𝐺𝑤𝑜𝑟𝑘
State 2
State 1
-
-
+
-
+
-
-
+
-
+
+
+
-
+
+
-
+
-
-
+
-
+
Salt water
+
+
-
-
-
-
-
+
+
+
+
-
-
+
-
Salt water
𝑠𝑎𝑙𝑡 𝑤𝑎𝑡𝑒𝑟
𝑠𝑎𝑙𝑡 𝑤𝑎𝑡𝑒𝑟
𝑠𝑎𝑙𝑡 𝑤𝑎𝑡𝑒𝑟
𝐺𝑆𝑡𝑎𝑡𝑒2
− 𝐺𝑆𝑡𝑎𝑡𝑒1
= ∆𝐺𝑤𝑜𝑟𝑘
*
-
+
-
+
+
-
+
-
-
+
+
+
+
-
-
+
+
+
-
+
+
-
+
+
+
-
-
+
-
+
+
+
-
-
+
-
+
+
Work
-
+
-
+
-
+
-
+
-
-
+
+
-
+
+
+
-
-
-
-
-
+
-
-
+
-
-
+
+
+
-
-
+
-
+
+
+
-
+
-
+
-
+
+
-
State 2
State 1
-
-
+
-
+
-
-
+
+
+
+
-
+
+
-
+
-
+
+
-
+
-
Work
+
-
+
+
-
-
+
+
+
-
-
+
-
+
+
+
+
-
-
+
+
+
+
-
-
+
+
+
-
+
+
-
+
-
𝐺𝑆𝑡𝑎𝑡𝑒2 − 𝐺𝑆𝑡𝑎𝑡𝑒1 = ∆𝐺 𝐸𝑥𝑐𝑒𝑠𝑠
*
-
-
+
+
+
-
-
+
-
+
+
+
-
-
+
-
+
-
+
-
-
+
+
-
+
+
-
+
+
+
-
-
-
-
-
+
-
-
+
-
-
+
+
+
-
-
+
-
+
+
+
-
-
+
-
+
-
+
+
-
Ions are distributed in water. Simplify the system with a continuous dielectric medium with
a swarm of charge no discrete charges
Electrostatic
potential r
-
-
+
-
+
-
-
+
+
-
-
+
+
+
*
+
+
-
-
+
+
-
-
-
+
+
-
+
-
r
-
+
+
+
+
-
-
+
+
+
-
-
+
Reference ion
-
+
+
+
-
-
+
-
+
+
+
-
-
+
Volume
element dV
+
+
-
Distance r
from
central ion
+
Ion cloud with
charge density
r
*
Images that explain physically why activity
coefficients are needed
Resistance of electrons to
move at a fixed potential
Material
Resistance of electric field to
transmit through substance
Resistivity, R
@20C ρ (Ω·m)
Copper
10-8
Carbon steel
(1010)
10-7
Sea water
Drinking water
Deionized water
Glass
Hard Rubber
𝑅(𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒) =
10
-1
10
3
10
5
10
12
10
13
Material
More
Insulating
of current
𝑉(𝑣𝑜𝑙𝑡𝑎𝑔𝑒)
𝐼 (𝑐𝑢𝑟𝑟𝑒𝑛𝑡)
*
Dielectric
constant, Dc
@20-25C
Air
1
Hard Rubber
3
Asbestos
4.8
Olive oil
3.1
Acetone
21
Ethanol
24
Ethylene glycol
37
Deionized water
80
More
shielding
of electric
field
𝑞∗𝑄
𝑟2
𝑡𝑒𝑠𝑡 𝑐ℎ𝑎𝑟𝑔𝑒 ∗ 𝑓𝑖𝑒𝑙𝑑 𝑐ℎ𝑎𝑟𝑔𝑒
= 𝑝𝑒𝑟𝑚𝑖𝑡𝑡𝑖𝑣𝑖𝑡𝑦 ∗
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑞 𝑎𝑛𝑑 𝑄
𝐹 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑓𝑜𝑟𝑐𝑒, 𝑣𝑒𝑐𝑡𝑜𝑟 = 𝑘𝑒
Polarized H2O reacting to an Electric field
Attenuated flux
𝐸′
𝑥,𝑦,𝑧
Electric field flux
Electron density
shifts towards field
𝑬
𝑥,𝑦,𝑧
+
+
+
+
+
electrons shift towards
the positive charge,
creating an internal
electric field EH2O
A dielectric material is one that polarizes in the presence of an electric field
*
A metal ion is a small “point charge” and produces an electric field
H2O molecules orient “coordinate” around it
0.7 Å
3.5 Å.
Slice it in half…
M+z
M+z
M+z is relatively small in size, only six water
can fit around it (in a sphere of course).
*
Layer
Na+
*
1
2
3
4
5
6
7
8
9
Inner
layer
radius
Å
0.8
3.5
6.2
8.9
11.6
14.3
17.0
19.7
22.4
Outer
layer
Layer
radius volume
Å
nm3
3.5
0.2
6.2
0.8
8.9
1.9
11.6
3.6
14.3
5.7
17.0
8.3
19.7
11.4
22.4
15.0
25.1
19.1
H2O in
layer
N
6
27
64
119
189
276
380
500
636
H2O in
sphere
N
6
33
97
216
405
681
1061
1561
2197
*
Na+
distance
Cl-
*



Start with 1m3 deionized water
Add 0.1 mol NaCl
Calculate the ion population density
𝑚𝑜𝑙𝑒 𝑁𝑎𝐶𝑙
𝑚𝑜𝑙𝑒 𝑖𝑜𝑛
𝑖𝑜𝑛
1𝑚3
23
∗
2
∗
6.023𝑥10
∗
𝑚3
𝑚𝑜𝑙𝑒 𝑁𝑎𝐶𝑙
𝑚𝑜𝑙𝑒 𝑖𝑜𝑛 1027 𝑛𝑚3
𝑖𝑜𝑛
= 1.20𝑥10−7
𝑛𝑚3
0.1

Invert to obtain the volume per ion
1.20𝑥10−7

𝑖𝑜𝑛
𝑛𝑚3
6
→
8.30𝑥10
𝑛𝑚3
𝑖𝑜𝑛
Calculate spherical radius of this volume
6
8.30𝑥10
𝑟=
*
4∗𝜋
𝑛𝑚3
𝑖𝑜𝑛
3
1
3
= 125 𝑛𝑚
NaCl conc, NaCl conc, Inter-ion
H2O
Act Coef,
mol/m3
mol/l
distance, molecules

nm
per ion
0
0


1
1e-6
1e-9
579
2.78e10
1.000
1e-5
1e-8
270
2.78e9
1.000
1e-4
1e-7
125
2.78e8
1.000
1e-3
1e-6
58
2.78e7
0.999
1e-2
1e-5
27
2.78e6
0.996
1e-1
1e-4
12.5
2.78e5
0.988
1
0.001
5.8
27775
0.965
10
0.01
2.7
2778
0.902
100
1000
0.1
1
1.25
0.58
278
28
0.771
0.644
*
*
* Ions can be visualized as point charge surrounded by water
* Water shields the electric fields created by each ion
* In dilute solutions, the ions are too far apart for the electric fields to
overlap
Distance=250 nm
Na+
r=125 nm
r=125 nm
There is no interaction
between the two ions,
they are too far apart
Sr+2
=the individual ion
*
=the water sheath or the water oriented around the ion
δ- =partial charge around the water sheath
Cl-
When concentration increases, ions distance decreases. Electric
fields start to overlap other ions and the water surrounding them
δ+
δ+
δ+
δ+
Sr+2
δ+
δ+
Sr+2
δ+
δ+
δ+
δ+
Electric fields start to
overlap
δ+
δ-
δ-
δ-
This changes
ion behavior
δ-
δ-
δ+
δ-
δ-
δδSO4-2
δδ-
SO4-2
δ-
*
* This electric field (electrostatic) effect is quantified using an
activity coefficient :

z
i  e
2
i A
I

δ+
δ+
δ+
Sr+2
Where:
δ+
+
δ+ δ
δ+
The charges are
starting to interact
δ-
1
I  ionic strength   z i2 mi
2
mi  concentration of ion
δ+
Sr+2
+ +
δ+ δ δ
zi  charge on ion
A  constant
δ+
δ-
δ- δ -
δ-
δ
δδ-
-
SO4-2
SO4-2
δ-
This is called the Debye Huckel model
*
δ-
δ-
δ-




Most common model for estimating activity coefficients
Assumes salts are completely ionized (limiting law)
Valid up to 0.01 mole/L salinity
Equation is -logγ± = Az+z-√I

where:
o γ± = mean of the activity coefficients for the + and - ions;
o A is a constant that depends on temperature and the dielectric
constant ε (A = 1.825 x 106(εT)-3/2 = 0.51 at 25◦ C in water);
o
o
z+ and z- are the + and - ion charges
I is the ionic strength.
*
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*Activity coefficients becomes part of the equilibrium
equation
𝐴+𝐵 =𝐶
𝛾𝐶 ∗ 𝐶
𝐾=
𝛾𝐴 ∗ 𝐴 ∗ 𝛾𝑏 ∗ 𝐵
*In dilute solutions, 1 , and
𝐶
𝐾=
𝐴 ∗ 𝐵
*
Ionic strength is a measure of the system’s
electric field.
𝐼
1
=
2
𝑚𝑖 𝑧𝑖 2
Where
𝑖
mi = ion concentration & zi = ion charge
Example
A 1.0 mol/kg solution of NaCl has 1.0 moles of Na+1 ion and 1.0 moles of
Cl-1 ion per Kg H2O.
I


1
2
2

Z Na 1  mNa 1   Z Cl 1  mCl 1 
2
I

The ionic strength is 1.0 molal.
*

1 2
1 1.0   12 1.0  1.0
2
A 1.0 mol/kg solution of CaSO4 has 1.0 moles of Ca+2 ion and
1.0 moles of SO4-2 ion per Kg H2O.


1
2
2
I  Z Ca 2  mCa 2   Z SO 42  mSO 42 
2


1
2
2
I  2  1.0    2  1.0   4.0
2
The ionic strength is 4.0 mol
*

Debye-Huckel model (1922), most common and easiest
ln  i    zi2 A I


Good to about 0.01 mol/L concentration
Extended Debye-Huckel model – extends the concentration limits


I

ln  i    z A
 bI 
1 I

Good to about 0.1 mol/L
2
i


Davies Equation (1938) – a further extension

Good to about 0.3 mol/L
*
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 Pitzer Equation
 2   
ln    | Z  Z  | f  m



1.5
 

 
2 2   
 B  m 
C 







 Bromley-Zematis
 A | Z Z | Equation
I 0.06  0.6 B  | Z
Log   

1 I
 Helgeson
 A | Z Z
Log   

i
l
| I
1  a0 B I


Z | I

1.5 
1 

|
Z
Z
|
I
 



    k
k

2
 i ,k
b
Y
I

k k k 
k
 BI  CI 2  DI 3
bil Yl I  l ,k
l   
l
k
*
All are good to between 10 and 30 mol/kg H2O. The problem is trying to use them…
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bil Yi I 
i  
i

 To get better activity coefficients, we need the more
complex models
 Complex models cannot be solved by hand (they are
non-linear and need multiple iterations)
 Thus freeware and commercial software products
were developed
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